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mathematics
calculus with applications
Calculus For Business, Economics And The Social And Life Sciences 11th Brief Edition Laurence Hoffmann, Gerald Bradley, David Sobecki, Michael Price - Solutions
A manufacturer with exclusive rights to a sophisticated new industrial machine is planning to sell a limited number of the machines to both foreign and domestic firms. The price the manufacturer can expect to receive for the machines will depend on the number of machines made available. It is
In Exercises 37 through 44, sketch the region of integration for the given integral and set up an equivalent integral with the order of integration reversed. Vx f(x, y) dy dx
A manufacturer is planning to sell a new product at the price of A dollars per unit and estimates that if x thousand dollars is spent on development and y thousand dollars on promotion, consumers will buy approximatelyunits of the product. If manufacturing costs are $50 per unit, express profit in
In Exercises 35 through 40, use the chain rule to find dz/dt. Express your answer in terms of x, y, and t. 3x z = =; x = t, y = P y
In Exercises 37 and 38, you will need to know that a closed cylinder of radius R and length L has volume V = πR2L and surface area S = 2πRL + 2πR2. The volume of a hemisphere of radius R is V = 2/3πR3 and its surface area is S = 2πR2.A bacterium is shaped like a cylindrical rod. If the volume
In Exercises 37 through 44, sketch the region of integration for the given integral and set up an equivalent integral with the order of integration reversed. 2y SS 0 f(x, y) dx dy
In Exercises 35 through 40, use the chain rule to find dz/dt. Express your answer in terms of x, y, and t.z = 2x + 3y; x = t2, y = 5t
In economics, the marginal product of labor is the rate at which output Q changes with respect to labor L for a fixed level of capital investment K. An economic law states that under certain circumstances, the marginal product of labor increases as the level of capital investment increases.
In Exercises 37 through 44, sketch the region of integration for the given integral and set up an equivalent integral with the order of integration reversed. 5.1² Jy/2 f(x, y) dx dy
In Exercises 35 through 40, use the chain rule to find dz/dt. Express your answer in terms of x, y, and t.z = x2y; x = 3t + 1, y = t2 − 1
Using x skilled workers and y unskilled workers, a manufacturer can produce Q(x, y) = 60x1/3y2/3 units per day. Currently the manufacturer employs 10 skilled workers and 40 unskilled workers and is planning to hire 1 additional skilled worker. Use calculus to estimate the corresponding change that
Suppose the manufacturer in Exercise 40 has only $11,000 to spend on the development and promotion of the new product. How should this money be allocated to generate the largest possible profit?Data from Exercises 40A manufacturer is planning to sell a new product at the price of $350 per unit and
Suppose that when x machines and y worker-hours are used each day, a certain factory will produce Q(x, y) = 10xy cell phones. Describe the relationship between the inputs x and y that results in an output of 1,000 phones each day.
In Exercises 37 through 44, sketch the region of integration for the given integral and set up an equivalent integral with the order of integration reversed. 1 2 f(x, y) dy dx In x
In Exercises 40 through 49, assume that the required extreme value is a relative extremum.The social desirability of an enterprise often involves making a choice between the commercial advantage of the enterprise and the social or ecological loss that may result. For instance, the lumber industry
In Exercises 35 through 40, use the chain rule to find dz/dt. Express your answer in terms of x, y, and t. Z x + y x - y x = ³+1, y = 1 - ²
In Exercises 37 through 44, sketch the region of integration for the given integral and set up an equivalent integral with the order of integration reversed. In 3 /3 Jo f(x, y) dy dx
In Exercises 37 through 44, sketch the region of integration for the given integral and set up an equivalent integral with the order of integration reversed. f(x, y) dy dx J-1Jx²+1
In Exercises 40 through 49, assume that the required extreme value is a relative extremum.Consider an experiment in which a subject performs a task while being exposed to two different stimuli (for example, sound and light). For low levels of the stimuli, the subject’s performance might actually
Suppose the utility derived by a consumer from x units of one commodity and y units of a second commodity is given by the utility function U(x, y) = 2x3y2. The consumer currently owns x = 5 units of the first commodity and y = 4 units of the second. Find the consumer’s current level of utility,
Using x skilled and y unskilled workers, a manufacturer can produce Q(x, y) = 3x + 2y units per day. Currently the workforce consists of 10 skilled workers and 20 unskilled workers.a. Compute the current daily output.b. Find an equation relating the levels of skilled and unskilled labor if the
In Exercises 35 through 40, use the chain rule to find dz/dt. Express your answer in terms of x, y, and t.z = x1/2y1/3; x = 2t, y = 2t2
Letwhere x > 0, y > 0. How do you know that f must have a minimum in the region x > 0, y > 0? Find the minimum. f(x, y) 12 X + 18 y + xy,
Use the method of Lagrange multipliers to prove that of all rectangles with a given perimeter, the square has the largest area.
Suppose a loan of A dollars is amortized over n years at an annual interest rate r compounded monthly. Let i = r/12 be the equivalent monthly rate of interest. Then the monthly payments will be M dollars, wherea. Allison has a home mortgage of $250,000 at the fixed rate of 5.2% per year for 15
The annual productivity of a certain country isunits, where K is capital expenditure in millions of dollars and L measures the labor force in thousands of worker-hours.a. Find the marginal productivity of capital and the marginal productivity of labor.b. Currently, capital expenditure is 5.041
In Exercises 41 through 46, the demand functions for a pair of commodities are given. Use partial derivatives to determine whether the commodities are substitute, complementary, or neither. D₁ = 1,000 0.02p1 -0.05p2; D₂ = 800 -0.001p² - P₁P2 D2
In Exercises 35 through 40, use the chain rule to find dz/dt. Express your answer in terms of x, y, and t.z = xy; x = e2t, y = e−3t
Suppose a company requires N units per year of a certain commodity. Suppose further that it costs D dollars to order a shipment of the commodity and that the storage cost per unit is S dollars per year. The units are used (or sold) at a constant rate throughout the year, and each shipment is used
Suppose the manufacturer in Exercise 41 decides to spend $12,000 instead of $11,000 on the development and promotion of the new product. Use the Lagrange multiplier λ to estimate how this change will affect the maximum possible profit.Data from Exercises 41Suppose the manufacturer in Exercise 40
In Exercises 41 through 46, the demand functions for a pair of commodities are given. Use partial derivatives to determine whether the commodities are substitute, complementary, or neither. D₁ = 3,000 + 400 P₁ + 3 + 50p2; D2 = 2,000 100p1 + 500 P2 + 4
A constant elasticity of substitution (CES) production function is one with the general formwhere K is capital expenditure; L is the level of labor; and A, α, β are constants that satisfy A > 0, 0 −1. Show that such a function has constant returns to scale: that is, for any constant
In Exercises 41 through 46, the demand functions for a pair of commodities are given. Use partial derivatives to determine whether the commodities are substitute, complementary, or neither. D₁ 7p2 1 + pi D₂ P1 1 + P₂
A farmer wishes to fence off a rectangular pasture along the bank of a river. The area of the pasture is to be 3,200 square meters, and no fencing is needed along the river bank. Find the dimensions of the pasture that will require the least amount of fencing.
In Exercises 41 through 46, the demand functions for a pair of commodities are given. Use partial derivatives to determine whether the commodities are substitute, complementary, or neither.D1 = 500 − 6p1 + 5p2; D2 = 200 + 2p1 − 5p2
Use Lagrange multipliers to find the possible maximum or minimum points on that part of the surface z = x − y for which y = x5 + x − 2. Then use your calculator to sketch the curve y = x5 + x − 2 and the level curves to the surface f(x, y) = x − y and show that the points you have just
In Exercises 40 through 49, assume that the required extreme value is a relative extremum.In relation to a rectangular map grid, four oil rigs are located at the points (−300, 0), (−100, 500), (0, 0), and (400, 300) where units are in feet. Where should a maintenance shed M(a, b) be located to
Suppose output Q is given by the Cobb-Douglas production function Q(K, L) = AKαL1−α, where A and α are positive constants and 0 < α < 1. Show that if K and L are both multiplied by the same positive number m, then the output Q will also be multiplied by m; that is, show that Q(mK,
In Exercises 40 through 49, assume that the required extreme value is a relative extremum.Alternative forms of a gene are called alleles. Three alleles, designated A, B, and O, determine the four human blood types A, B, O, and AB. Suppose that p, q, and r are the proportions of A, B, and O in a
In Exercises 54 through 57, find the partial derivatives fx and fy and then use your graphing utility to determine the critical points of each function. f(x, y) x² + xy + 7y² x ln y
Repeat Exercise 56 for the functionData from Exercises 56Let F(x, y) = x2 + 2xy − y2.a. If F(x, y) = k for constant k, use the method of implicit differentiation developed in Chapter 2 to find.b. Find the partial derivatives Fx and Fy and verify that F(x, y) = xex + y X + x ln(x + y)
In Exercises 40 through 49, assume that the required extreme value is a relative extremum.Four small towns in a rural area wish to pool their resources to build a television station. If the towns are located at the points (−5, 0), (1, 7), (9, 0), and (0, −8) on a rectangular map grid, where
A storage shed is to be constructed of material that costs $15 per square foot for the roof, $12 per square foot for the two sides and back, and $20 per square foot for the front. What are the dimensions of the largest shed (in volume) that can be constructed for $8,000?
In Exercises 45 through 54, use a double integral to find the area of R.R is the region in the first quadrant bounded by y = 4 − x2, y = 3x, and y = 0.
In Exercises 52 and 53, evaluate the given double integral for the specified region R.where R is the rectangular region bounded by x = 0, x = 1, y = −2, and y = 2 1x JR (x + 2y) dA,
The lens equation in optics says thatwhere do is the distance of an object from a thin, spherical lens; di is the distance of its image on the other side of the lens; and F is the focal length of the lens (see the accompanying figure). Express F in terms of do and di, and sketch several level
Let F(x, y) = x2 + 2xy − y2.a. If F(x, y) = k for constant k, use the method of implicit differentiation developed in Chapter 2 to find.b. Find the partial derivatives Fx and Fy and verify that dy dx || Fx Fy
Van der Waal’s equation of state says that 1 mole of a confined gas satisfies the equationwhere T (ºC) is the temperature of the gas, V (cm3) is its volume, P (atmospheres) is the pressure of the gas on the walls of its container, and a and b are constants that depend on the nature of the gas.a.
Two competing brands of power lawnmowers are sold in the same town. The price of the first brand is x dollars per mower, and the price of the second brand is y dollars per mower. The local demand for the first brand of mower is given by a function D(x, y).a. How would you expect the demand for the
It is estimated that the weekly output at a certain plant is given by Q(x, y) = 1,175x + 483y + 3.1x2y − 1.2x3 − 2.7y2 units, where x is the number of skilled workers and y is the number of unskilled workers employed at the plant. Currently the workforce consists of 37 skilled and 71 unskilled
Let f (x, y) = x2 + y2 − 4xy. Show that f does not have a relative minimum at its critical point (0, 0), even though it does have a relative minimum at (0, 0) in both the x and y directions.
The monthly demand for a certain brand of toasters is given by a function f(x, y), where x is the amount of money (measured in units of $1,000) spent on advertising and y is the selling price (in dollars) of the toasters. Give economic interpretations of the partial derivatives fx and fy. Under
In Exercises 45 through 54, use a double integral to find the area of R.R is the region bounded by y = 16/x, y = x, and x = 8.
In manufacturing semiconductors, it is necessary to use water with an extremely low mineral content, and to separate water from contaminants, it is common practice to use a membrane process called reverse osmosis. A key to the effectiveness of such a process is the osmotic pressure, which may be
Suppose the output Q of a factory depends on the amount K of capital investment measured in units of $1,000 and on the size L of the labor force measured in worker-hours. Give an economic interpretation of the second-order partial derivative 2 не oze
The demand function for peanut butter iswhile that for a second commodity isIs the second commodity more likely to be jelly or bread? Explain. D₁(P₁, P2) = 800 0.03p²-0.04p2
At a certain factory, the output is Q = 120K1/2L1/3 units, where K denotes the capital investment measured in units of $1,000 and L the size of the labor force measured in worker-hours.a. Determine the sign of the second-order partial derivative ∂2Q/∂L2, and give an economic
The demand function for a certain brand of gel pens iswhile that of a second commodity isIs the second commodity more likely to be pencils or paper? Explain. D₁(P₁, P₂) = = 700 700 - 4pi + 7P₁P₂
In Exercises 55 through 64, find the volume of the solid under the surface z = f(x, y) and over the given region R.f(x, y) = 6 − 2x − 2y; R: 0 ≤ x ≤ 1, 0 ≤ y ≤ 2
Find the volume under the surface z = 2xy and above the rectangle with vertices (0, 0), (2, 0), (0, 3), and (2, 3).
The level curves in the land areas of the accompanying figure indicate ice elevations above sea level (in meters) during the last major ice age (approximately 18,000 years ago). The level curves in the sea areas indicate sea surface temperature. For instance, the ice was 1,000 meters thick above
In Exercises 54 through 57, find the partial derivatives fx and fy and then use your graphing utility to determine the critical points of each function.f(x, y) = (x2 + 3y − 5)e−x2−2y2
Find the volume under the surface z = xe−y and above the rectangle bounded by the lines x = 1, x = 2, y = 2, and y = 3.
Sometimes you can classify the critical points of a function by inspecting its level curves. In each of the following cases, determine the nature of the critical point of f at (0, 0).a.b. 1=3²/33 두=1 l=f
In Exercises 54 through 57, find the partial derivatives fx and fy and then use your graphing utility to determine the critical points of each function.f(x, y) = 2x4 + y4 − x2(11y − 18)
Plot the points (1, 1), (1, 2), (3, 2), and (4, 3), and use partial derivatives to find the corresponding least-squares line.
In Exercises 58 through 61, use the method of Lagrange multipliers to find the indicated maximum or minimum. You will need to use the graphing utility or the solve application on your calculator.Maximize f(x, y) = xex2−y subject to x2 + 2y2 = 1.
In Exercises 55 through 64, find the volume of the solid under the surface z = f(x, y) and over the given region R.f(x, y) = 9 − x2 − y2; R: −1 ≤ x ≤ 1, −2 ≤ y ≤ 2
In Exercises 55 through 64, find the volume of the solid under the surface z = f(x, y) and over the given region R.f(x, y) = 1/xy;R: 1 ≤ x ≤ 2, 1 ≤ y ≤ 3
Find the average value of f(x, y) = xy2 over the rectangular region with vertices (−1, 3), (−1, 5), (2, 3), and (2, 5).
In Exercises 58 through 61, use the method of Lagrange multipliers to find the indicated maximum or minimum. You will need to use the graphing utility or the solve application on your calculator.Minimize f(x, y) = ln(x + 2y) subject to xy + y = 5.
Find three positive numbers x, y, and z so that x + y + z = 20 and the product P = xyz is a maximum.
In Exercises 54 through 57, find the partial derivatives fx and fy and then use your graphing utility to determine the critical points of each function.f(x, y) = 6x2 + 12xy + y4 + x − 16y − 3
In Exercises 58 through 61, use the method of Lagrange multipliers to find the indicated maximum or minimum. You will need to use the graphing utility or the solve application on your calculator.Minimizesubject to x + 2y = 7. f(x,y) 3 1 + 2 ху
In Exercises 58 through 61, use the method of Lagrange multipliers to find the indicated maximum or minimum. You will need to use the graphing utility or the solve application on your calculator.Maximize f(x, y) = ex+y − x ln (x/y) subject to x + y = 4.
Find three positive numbers x, y, and z so that 2x + 3y + z = 60 and the sum S = x2 + y2 + z2 is minimized.
Each of Exercises 61 through 68 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.The demand for a certain product is Q(x, y) = 200 − 10x2 + 20xy units per month, where x is the price of the product and y is the price of
The marketing manager for a certain company has compiled these data relating monthly advertising expenditure and monthly sales (both measured in units of $1,000):a. Plot these data on a graph.b. Find the least-squares line, and add it to the graph in part (a).c. Use the least-squares line to
Repeat Exercise 59 with the output function Q(x, y) = 1,731x + 925y + x2y − 2.7x2 − 1.3y3/2 and initial employment levels of x = 43 and y = 85.Data from Exercises 59It is estimated that the weekly output at a certain plant is given by Q(x, y) = 1,175x + 483y + 3.1x2y − 1.2x3 − 2.7y2 units,
In Exercises 55 through 64, find the volume of the solid under the surface z = f(x, y) and over the given region R.f(x, y) = ex+y; R: 0 ≤ x ≤ 1, 0 ≤ y ≤ In 2
Suppose the daily output Q of a factory depends on the amount K of capital investment and on the size L of the labor force. A law of diminishing returns states that in certain circumstances, there is a value L0 such that the marginal product of labor will be increasing for L < L0 and decreasing
In Exercises 55 through 64, find the volume of the solid under the surface z = f(x, y) and over the given region R.f(x, y) = xe−y; R: 0 ≤ x ≤ 1, 0 ≤ y ≤ 2
Find the shortest distance from the origin to the surface y2 − z2 = 10.
In Exercises 55 through 64, find the volume of the solid under the surface z = f(x, y) and over the given region R.f(x, y) = (1 − x)(4 − y); R: 0 ≤ x ≤ 1, 0 y ≤ 4
In Exercises 55 through 64, find the volume of the solid under the surface z = f(x, y) and over the given region R.f(x, y) = ey2; R is bounded by x = 2y, x = 0, and y = 1.
In Exercises 55 through 64, find the volume of the solid under the surface z = f(x, y) and over the given region R.f(x, y) = 2x + y; R is bounded by y = x, y = 2 − x, and y = 0.
Each of Exercises 61 through 68 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.Using x hours of skilled labor and y hours of unskilled labor, a manufacturer can produce Q(x, y) = 10xy1/2 units. Currently 30 hours of
Suppose the utility derived by a consumer from x units of one commodity and y units of a second commodity is given by the utility function U(x, y) = x3y2. The consumer currently owns x = 5 units of the first commodity and y = 4 units of the second. Use calculus to estimate how many units of the
Each of Exercises 61 through 68 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.A car dealer determines that if gasoline-electric hybrid automobiles are sold for x dollars apiece and the price of gasoline is y cents per
In Exercises 55 through 64, find the volume of the solid under the surface z = f(x, y) and over the given region R.f(x, y) = x + 1; R is bounded by y = 8 − x2 and y = x2.
A paint company makes two brands of latex paint. Sales figures indicate that if the first brand is sold for x dollars per quart and the second for y dollars per quart, the demand for the first brand will be Q quarts per month, where Q(x, y) = 200 + 10x2 − 20y It is estimated that t months from
In Exercises 55 through 64, find the volume of the solid under the surface z = f(x, y) and over the given region R.f(x, y) = 4xey; R is bounded by y = 2x, y = 2, and x = 0.
Each of Exercises 61 through 68 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.Ethan is an investor who derives U(x, y) units of satisfaction from owning x stock units and y bond units, whereHe currently owns x = 27
Recall from Exercise 47 that the surface area of a person’s body may be measured by the empirical formula S(W, H) = 0.0072W0.425 H0.725 where W (kg) and H (cm) are the person’s weight and height, respectively. Currently, a child weighs 34 kg and is 120 cm tall.a. Compute the partial derivatives
Each of Exercises 61 through 68 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.A grocer’s daily profit from the sale of two brands of flavored iced tea iscents, where x is the price per bottle of the first brand and y
The difference between an animal’s surface temperature and that of the surrounding air causes a transfer of energy by convection. The coefficient of convection h is given bywhere V (cm/sec) is wind velocity, D(cm) is the diameter of the animal’s body, and k is a constant.a. Find the partial
Suppose that when apples sell for x cents per pound and bakers earn y dollars per hour, the price of apple pies at a certain supermarket chain isdollars per pie. Suppose also that t months from now, the price of apples will be x = 129 − √8t cents per pound and bakers’ wages will be y = 15.60
Arnold, the heat-seeking mussel, is the world’s smartest mollusk. Arnold likes to stay warm, and by using the crustacean coordinate system he learned from a passing crab, he has determined that at each nearby point (x, y) on the ocean floor the temperature (ºC) is T(x, y) = 2x2 − xy + y2 −
Nuclear waste is often disposed of by sealing it into containers that are then dumped into the ocean. It is important to dump the containers into water shallow enough to ensure that they do not break when they hit bottom. Suppose as the container falls through the water, there is a drag force that
Each of Exercises 61 through 68 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.At a certain factory, when the capital expenditure is K thousand dollars and L worker-hours of labor are employed, the daily output will be
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